A hybridizable discontinuous Galerkin method on unfitted meshes for single-phase Darcy flow in fractured porous media. (March 2023) - Fu, Guosheng; Yang, Yang
Abstract: We present a novel hybridizable discontinuous Galerkin (HDG) method on unfitted meshes for single-phase Darcy flow in a fractured porous medium. In particular we apply the HDG methodology to the recently introduced reinterpreted discrete fracture model (RDFM) that use Dirac- δ functions to model both conductive and blocking fractures. Due to the use of Dirac- δ function approach for the fractures, our numerical scheme naturally allows for unfitted meshes with respect to the fractures, which is the major novelty of the proposed scheme. Moreover, the scheme is locally mass conservative. In particular, our scheme has a simple form, which is a novel modification of an existing regular Darcy flow HDG solver by adding the following two components: (i) locate the co-dimension one fractures in the background mesh and add the appropriate surface integrals associated with these fractures into the stiffness matrix, (ii) adjust the penalty parameters on cells cut through conductive and blocking fractures (fractured cells). Despite the simplicity of the proposed scheme, it performs quite well for various benchmark test cases in both two and three dimensions. To the best of our knowledge, this is the first time that an unfitted finite element scheme been applied to complex fractured porous media flow problems in 3D with both blocking and conductive fractures without any restrictions on the meshes. Highlights: Introduce line Dirac delta functions into the discrete fracture model for fracture and barrier networks. The model works for both blocking and conductive fractures without any restrictions on the meshes. The hybridizable discontinuous Galerkin methods for the model is conservative and easy to implement. Prove the well-posedness of the proposed scheme.